Identification of an obstruction into a fluid duct, divergence free radial basis functions, and applications
Dr. Pedro González CasanovaInstituto de Matemáticas, Universidad Nacional Autónoma de México
The problem of identifying an obstruction into a fluid duct has several major applications, for example in medicine the presence of a stenosis in a coronary vessels is a life threaten disease. In this talk we formulate a continuous setting and study from a numerical perspective the inverse problem of identifying an obstruction contained in a 2D elastic duct where a Stokes flow hits the boundary (Navier--slip boundary conditions), generating an acoustic waves. To be precise, by using acoustic wave measurements at certain points at the exterior to the duct, we are able to identify the location; extension and height of the obstruction. Thus, our framework constitutes an external approach for solving this obstacle inverse problem. A radial basis function approach based in Local Hermite Interpolation and hybrid kernels is introduced. Synthetic examples are used in order to verify the effectiveness of the proposed numerical formulation. A comparison between the RBFs method and the Finite element technique is presented. This work was done in collaboration with: L. Breton, J. López Estrada and C. Montoya
SHORT BIO
Dr. Pedro Gonzalez Casanova did his PhD in mathematics at the University of Oxford, U. K. He was also aresearch fellow for one year at the University of Oxford. He was a visiting scholar, for one year, at the University Paul Sabatier in Toulouse, France. His research focuses on partial differential equations, inverse problems, and control problems, both from a theoretical and numerical perspective. He developed and appliedGodunov type methods for hyperbolic systems (relativistic flows). A major interest of Pedro's is the formulation, analysis, and application of mesh free radial basis function methods to direct, inverse, and control PDE problems. He coordinated for more than ten years a research group in the field of numerical solution of PDEs by Radial Basis Functions, with the participation of different universities, research centers, and researchers from Mexico and France. As a result of this project, several articles were published and several undergraduate and PhD students have obtained their degrees. Pedro has published more than 30 research articles, he has presented more than 140 talks in both Mexican and international congresses and conferences; and he has organized both national and international congress and workshops in applied mathematics. He has also been director of several PhD and undergraduate theses.and a referee for several international journals in numerical analysis. He is a reviewer for the MathematicalReviews of the American Mathematical Society. Currently, he is a researcher at the Institute of Mathematics of the National Autonomous University of Mexico.